The Equation of a Circle

The Equation of a Circle — Moosa Academy

A circle is the simplest of the conic sections: every point on it sits exactly the same distance from a single fixed centre. That one sentence is enough to derive its entire equation — and the equation, once you can read it, tells you the centre and radius at a glance.

Concept The definition and the three key lengths

Centre — the fixed point every point on the circle is measured from.

Radius

r

— that fixed distance.

Diameter — the longest chord, passing through the centre; always

2r

Concept The standard form

(x - h)^2 + (y - k)^2 = r^2

Here $(h,\ k)$ is the centre and $r$ the radius. This is nothing more than the distance formula rearranged: the distance from $(x,\ y)$ to $(h,\ k)$ equals $r$ , squared on both sides to clear the square root.

Centred at the origin, this collapses to the simplest possible form:

x^2 + y^2 = r^2

With radius 1 as well, it becomes the unit circle,

x^2 + y^2 = 1

— the foundation of trigonometry.

Watch the signs carefully. $(x - 3)^2$ means $h = 3$ , but $(x + 3)^2$ is really $(x - (-3))^2$ , so $h = -3$ . The right-hand side is $r^2$ , not $r$ — a circle written as $\dots = 25$ has radius 5, not 25.

Example Writing the equation from centre and radius

Find the equation of the circle with centre $(-1,\ 4)$ and radius 6.

Substitute

h = -1

k = 4

r = 6

into the standard form.

(x - (-1))^2 + (y - 4)^2 = 6^2

⟹ (x + 1)² + (y − 4)² = 36

The negative $h$ turned into a plus inside the bracket — the single most common place to slip up.

Example Reading centre and radius from an equation

Find the centre, radius and diameter of $(x - 2)^2 + (y + 3)^2 = 25$ .

(x - 2)

gives

h = 2

;

(y + 3)

(y - (-3))

, so

k = -3

r^2 = 25 \Rightarrow r = 5

Diameter

= 2r = 10

⟹ centre (2, −3), radius 5, diameter 10

Concept The general form, and completing the square

x^2 + y^2 + Dx + Ey + F = 0

Expanding the standard form always produces an equation of this shape. Going the other way — from general back to standard — requires completing the square separately on the $x$ terms and the $y$ terms. Only then can you read off the centre and radius.

Example Converting general form to standard form

Find the centre and radius of $x^2 + y^2 - 6x + 8y - 11 = 0$ .

Group the variables and move the constant across:

(x^2 - 6x) + (y^2 + 8y) = 11

Halve the

x

coefficient and square it: half of −6 is −3, and

(-3)^2 = 9

Halve the

y

coefficient and square it: half of 8 is 4, and

4^2 = 16

Add both to each side:

(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16

Factor each bracket:

(x - 3)^2 + (y + 4)^2 = 36

⟹ centre (3, −4), radius 6

Whatever you add to complete a square on the left must be added to the right as well — forgetting this is what produces a wrong radius.

Note Properties worth remembering

Circumference

= 2\pi r

and area

= \pi r^2

A central angle is twice the inscribed angle standing on the same arc.

An angle inscribed in a semicircle is always a right angle.

Chords of equal length sit equally far from the centre.

A tangent meets the circle at exactly one point, and is perpendicular to the radius drawn to that point.

Note How a line can meet a circle

Two points — the line is a secant.

One point — the line is a tangent.

No points — the line misses the circle entirely.

Two circles behave similarly: they may miss each other, touch at one point (externally or internally), or cross at two points — decided entirely by the distance between their centres compared with their radii.

Note Mistakes to avoid

Reading the right-hand side as

r

rather than

r^2

Getting the centre's signs backwards:

(x + 5)^2

means

h = -5

Adding the completing-the-square terms to only one side of the equation.

Confusing radius with diameter when a problem gives you one and asks for the other.

Not checking that

r^2 > 0

— if it comes out negative, no real circle exists.

Summary

A circle is the set of points a fixed distance r from a centre (h, k).
Standard form (x − h)² + (y − k)² = r² shows the centre and radius directly; the right-hand side is r², not r.
General form x² + y² + Dx + Ey + F = 0 hides them — complete the square on x and y to recover the standard form.
Centred at the origin the equation is x² + y² = r², and with r = 1 it is the unit circle used throughout trigonometry.